Abstract: Using Boolean arithmetic, the index of convergence and period of a Boolean matrix A are respectively the least non-negative integer k and the least positive integer p respectively such that A^k=A^k+p. The knowledge about the period is quite completed. Given positive integers n and s satisfying a
certain inequality, the authors apply combinatorial arguments to determine
which numbers are attainable as the indices of convergence of
n× n Boolean matrices whose digraphs contain at least one s-cycle.

Key words: Boolean matrix, Index ofconvergence, Period, Digraph, Cycle